Identification in financial models with time-dependent volatility and stochastic drift components [Elektronische Ressource] / vorgelegt von Romy Krämer
142 Pages
English
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Identification in financial models with time-dependent volatility and stochastic drift components [Elektronische Ressource] / vorgelegt von Romy Krämer

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142 Pages
English

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Identification in Financial Models withTime-Dependent Volatility andStochastic Drift ComponentsDISSERTATIONzur Erlangung des akademischen GradesDoctor rerum naturalium(Dr. rer. nat.)TECHNISCHE UNIVERSITÄT CHEMNITZFakultät für Mathematikvorgelegt von Dipl.-Math. Romy Krämergeb. am 08.02.1980 in Karl-Marx-Stadt (Chemnitz)Betreuer: Prof. Dr. Bernd Hofmann (TU Chemnitz)Gutachter: Prof. Dr. Bernd Hofmann (TU Chemnitz)Dr. P. Mathé (WIAS Berlin)Prof. Dr. W. Grecksch (MLU Halle)Tag der Verteidigung: 31. Mai 2007Verfügbar im MONARCH der TU Chemnitz:http://archiv.tu-chemnitz.de/pub/2007/0080Contents1 Introduction 72 Stochastic Preliminaries 103 The Bivariate Ornstein-Uhlenbeck model 223.1 Solution of the stochastic differential equation . . . . . . . . . . . . . . . . . 243.2 Pricing of European Call Options . . . . . . . . . . . . . . . . . . . . . . . . 264 Volatility estimation by wavelet methods 284.1 Introduction to wavelet analysis . . . . . . . . . . . . . . . . . . . . . . . . . 284.2 The situation and the estimator . . . . . . . . . . . . . . . . . . . . . . . . . 344.3 Asymptotic study of the estimator . . . . . . . . . . . . . . . . . . . . . . . 364.3.1 Weak convergence of the estimator . . . . . . . . . . . . . . . . . . . 364.3.2 Mean integrated square error . . . . . . . . . . . . . . . . . . . . . . 464.4 Numerical case studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 584.4.

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Identification in Financial Models with
Time-Dependent Volatility and
Stochastic Drift Components
DISSERTATION
zur Erlangung des akademischen Grades
Doctor rerum naturalium
(Dr. rer. nat.)
TECHNISCHE UNIVERSITÄT CHEMNITZ
Fakultät für Mathematik
vorgelegt von Dipl.-Math. Romy Krämer
geb. am 08.02.1980 in Karl-Marx-Stadt (Chemnitz)
Betreuer: Prof. Dr. Bernd Hofmann (TU Chemnitz)
Gutachter: Prof. Dr. Bernd Hofmann (TU Chemnitz)
Dr. P. Mathé (WIAS Berlin)
Prof. Dr. W. Grecksch (MLU Halle)
Tag der Verteidigung: 31. Mai 2007
Verfügbar im MONARCH der TU Chemnitz:
http://archiv.tu-chemnitz.de/pub/2007/0080Contents
1 Introduction 7
2 Stochastic Preliminaries 10
3 The Bivariate Ornstein-Uhlenbeck model 22
3.1 Solution of the stochastic differential equation . . . . . . . . . . . . . . . . . 24
3.2 Pricing of European Call Options . . . . . . . . . . . . . . . . . . . . . . . . 26
4 Volatility estimation by wavelet methods 28
4.1 Introduction to wavelet analysis . . . . . . . . . . . . . . . . . . . . . . . . . 28
4.2 The situation and the estimator . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.3 Asymptotic study of the estimator . . . . . . . . . . . . . . . . . . . . . . . 36
4.3.1 Weak convergence of the estimator . . . . . . . . . . . . . . . . . . . 36
4.3.2 Mean integrated square error . . . . . . . . . . . . . . . . . . . . . . 46
4.4 Numerical case studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.4.1 The L-method as criteria for the choice of the resolution level . . . . 62
4.5 Outlook: Wavelet thresholding . . . . . . . . . . . . . . . . . . . . . . . . . . 68
5 Operator equations with Nemytskii operators 70
5.1 Inverse problems and regularization methods . . . . . . . . . . . . . . . . . . 70
5.2 Nemytskii operators: Acting conditions and continuity . . . . . . . . . . . . 76
5.3 Nemytskii operators with monotone generator functions . . . . . . . . . . . . 79
2CONTENTS 3
6 Identification of the time-dependent volatility using option prices 89
6.1 Inverse option pricing: Tikhonov-Regularization . . . . . . . . . . . . . . . . 89
6.2 The outer problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
6.2.1 Analytical studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
6.2.2 Numerical case studies concerning ill-conditioning effects . . . . . . . 102
6.2.3 Regularization by monotonization – Algorithm . . . . . . . . . . . . . 108
6.2.4 Discrete Variant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
7 Identification of the drift parameters 121
7.1 Preliminary considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
7.2 State space representation and Kalman filter . . . . . . . . . . . . . . . . . . 125
7.3 Likelihood function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
7.4 Numerical Case Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131Notations
Spaces
P polynomials of degree ≤nn
p p dL (I), L (I;R ) Lebesgue spaces of p-power integrable functions
pL (I) contains functions f :I →R, whereas
p d dL (I;R ) contains functions f :I →R
kC (I) space of k-times continuously differentiable functions f :I →R
If I = [a,b] we also writeC[a,b].
Furthermore, we use the abbreviationC when appropriate. P
2 2l (Z) := {x } :x ∈R and x <∞k k∈Z k k∈Z k P
2 2l (N) := {x } :x ∈R and x <∞k k∈N k kk∈N
sH (R) Sobolev spaces cf. (4.5)
<,> scalar product in the Hilbert space HH
Functions
[r] entier function ([r] is the largest integer that is less or equal than r)
1 s∈A
χ indicator function of the set A, i.e.χ (s) :=
A A 0 s∈/ A
Operators
R∞ −iξxˆFf(ξ) =f(ξ) := e f(x)dx : Fourier transform of f−∞
∗F adjoint operator of F
4CONTENTS 5
Norms
Pm,n 2 2|| For a matrix A = (a ) we set|A| := |a |i,j i,ji=1,j=1
i,j
kfk := max |f(x)|C(I) x∈I
1R
p p
pkfk := |f(x)| dxL (I) I
Miscellaneous
∧ r ∧r := min{r ,r }1 2 1 2
⊕ orthogonal sum
# cardinality
Vectors and Matrices
nI identity matrix, I = (δ )i,j i,j=1
0 vector that contains only zeros
Stochastics
Eξ Expectation of the random vector ξ
T2D ξ :=E(ξ−Eξ)(ξ−Eξ)
T
Cov(ξ,η) :=E(ξ−Eξ)(η−Eη)
A({ξ } ) The sigma-algebra generated by thec c∈C
random variables ξ (c∈C)c
ξ∼N(m,R) ξ is normally distributed with expectation m
and covariance matrix R
F (x) distribution function of the random vector ξξ R
i<t,x>φ (t) = e dF (x) characteristic function of the random vector ξξ n ξR
ξ =o (1) {ξ } converges to zero in probability, i.e.n P n
∀ε> 0 it holdsP(|ξ |>ε)→ 0 for n→∞.n
Equivalences
If A(u) and B(u) are positive functions of a set of parameters, the notation
A(u).B(u)6 CONTENTS
means, that there exists a constant C > 0 such that A(u) ≤ CB(u) independently of u.
Furthermore, the notation
A(u)hB(u)
means A(u). B(u) and B(u). A(u). Furthermore, we use the Landau symbols O and o
to describe the asymptotic behavior of functions. To be precise, for two functions f and g
we write
f(x) =O(g(x)) for x→∞
if and only if there exists an x and a constant M > 0 such that0
|f(x)|≤ M|g(x)| for x>x .0
Besides, we write
f(x) =o(g(x)) for x→∞
f(x)
if and only if → 0 for x→∞.
g(x)
In order to distinguish between results that are cited from the literature and own contribu-
tions we use the term proposition when we reformulate facts that are found in the literature.
As opposed to that lemmas and theorems state and prove assertions that we could not find
in the literature. As usual we use the term lemma for auxiliary results that are mainly used
to prove a theorem.Chapter 1
Introduction
During the last decades a great diversity of price models for financial assets has been de-
veloped. It is well-known that as long as it is only possible to observe asset prices (or the
corresponding returns) in a discrete scheme, it is always possible to find a model based on
a geometric Brownian motion with constant volatility coefficient and stochastic drift term
which has identical distributions as the observed returns (cf., e.g. [46]). Clearly, due to this
fact one must not argue that the empirically observed returns which fail to have indepen-
dent normal distributions require extensions of the classical model in order to price options
accurately.
On the other hand it is obvious that by introducing further random effects into the cor-
responding models via a drift for a given (fixed) behaviour of the observed data there are
changes in option prices, even though the option price formula itself is unaffected by changes
in the drift. Consequently, the study of corresponding models is meaningful. In this context
the estimate of volatility has to be reinterpreted in the light of the specific model which is
assumed.
Speaking generally, these models are based on stochastic processes which are specified by
several model parameters and these parameters have to be calibrated to observed market
data. Obviously,acorrectidentificationoftheseparametersisofcoreimportanceasotherwise
the models do not yield a good approximation of the real price processes. Moreover, the
model parameters are also necessary for pricing derivatives. A computation of these prices
with wrong parameters can lead to results which are far away from the prices observed on
real markets even if the correct formulas have been used.
Here and in what follows the term parameter means either a finite dimensional vector or a
function that specifies a model. This manner of speech is common in the literature of inverse
problems but it differs from the statistical literature. There, this term is generally used in
the meaning of a finite dimensional parameter vector. Consequently, the branch of statistics
which is concerned with the identification of finite dimensional parameter vectors is called
parametric statistics, whereas nonparametric statistics aims at the calibration of models
containing unknown functions (for example a volatility function). If we want to stress that
certain parameters are finite dimensional, we speak of finite dimensional parameter vectors
or real-valued parameters.
78 1. INTRODUCTION
The focus of this thesis is on parameter identification in market models with partial infor-
mation in which the stochastic drift of the logarithmic asset price process depends on an
unobservable state process. The asset price process is assumed to have a time-dependent but
deterministic volatility, which has to be identified. Furthermore, the stochastic drift and the
underlying stateprocess arecharacterised byafinite number ofreal-valued parameterswhich
are assumed tobeconstant with respect totime. The aim isan analysisofseveral calibration
techniques which are suitable for the identification of the described parameters. As a toy ex-
ample we consider a slightly modified version of the Bivariate Trending Ornstein-Uhlenbeck
model which has been introduced by Lo and Wang in [37].
Intheliteratureseveral calibrationtechniques arediscussed. Speakinggenerally, there areon
the one hand statistical approaches which aim at estimating the parameters from observed
asset prices. On the other hand there are approaches which use prices of observed derivatives
(e.g. observed option price data). In general, the last approach leads to inverse problems.
With respect to the calibration of a time-dependent volatility function there exist methods
of nonparametric statistics which use high-frequency asset price data. As an example we
consider the method of wavelets, which performs a projection on an orthonormal wavelet
basis (cf., e.g. [7], [16], [45, p. 268ff]). With this approach the volatility function can be
identified on the time interval on which asset prices are observed, i.e. on a time interval in
the past.
However, for pricing options and other derivatives the volatility function (or deviated quan-
tities) has to be known on a time interval [t ,t ] starting at the current time pointnow future
t and ending at some future time instant t . To calibrate the volatility over this timenow future
interval one can observe prices of options with maturities varying in [t ,t ]. In thisnow future
case the identification leads to the inverse problem of option pricing (cf., e.g. [9]), which is
known tobe ill-posed, i.e. the solution doesnot depend continuously on the data. In order to
overcome these ill-posedness effects many papers have been concerned with the applicability
of several regularization methods (cf., e.g. [11], [23], [27]).
Furthermore, with respect to optimisation of the utility from terminal wealth (cf., e.g. [34]
in the context of partial information) the necessity of a proper identification of all model
parameters is obvious. In other words, for utility optimisation the real-valued parameters
that characterise the stochastic drift terms have to be identified too. For a given volatility
function this can be done by maximum likelihood estimation (cf., e.g. [20], [36] and [39] for a
general introduction). Clearly, if the volatility input is itself a result of the above-mentioned
estimation methods, it becomes necessary to discuss the question which effects are caused by
small errors in this input.
This thesis combines the above-mentioned approaches. To keep notationsconsistent with the
literature it is inevitable to use the time interval [0,T] in different meanings. To be precise,
in the chapters that are concerned with statistical methods the notation [0,T] is used for
some time interval in the past. In this situation T denotes the current time point and 0 the
instant where the observation of the asset prices started. As opposed to that, in the context
of the inverse problem of option pricing the interval [0,T] denotes some future time interval
starting at the current time point t = 0.now
The thesis is organised as follows. In Chapter 2 we repeat some basic concepts and propo-
sitions from probability theory which are used at several places throughout the thesis. In1. INTRODUCTION 9
Chapter 3 we introduce a generalised Ornstein-Uhlenbeck model as a toy example for our
numerical case studies. After the presentation of the model and a short motivation we prove
unique solvability of the corresponding system of stochastic differential equations for the
logarithmic price-process and review results concerning option pricing in this model.
In Chapter 4 we present a nonparametric estimator of the squared volatility function which
performs a projection onto an orthonormal wavelet basis. We start with a short introduction
into wavelet analysis. After that we generalise convergence results for the considered estima-
tor to the situation of market models with incomplete information in which the stochastic
drift depends on an unobservable (possibly multidimensional) state process. Convergence is
studied in the weak sense as well as in terms of the mean integrated square error. Moreover,
forthe meanintegrated square erroraconvergence rateisproven. This rateisalsoillustrated
by means of a numerical case study. Furthermore, the data-driven choice of the resolution
level according to the L-method is discussed.
Chapter 5 is devoted to inverse problems that can be formulated as operator equations with
Nemytskii operators. After a general introduction into inverse problems and regularization
methods we review results about Nemytskii operators. Unfortunately, the literature is only
concerned with properties of the Nemytskii operators itself (such as acting conditions, con-
tinuity, differentiability) but does not address questions about existence and properties of
the corresponding inverse operators. Restricting our considerations to a certain type of Ne-
mytskii operators, namely Nemytskii operators generated by monotone functions we answer
some of these open questions in Section 5.3.
Chapter 6 addresses the calibration of the time-dependent volatility function σ (or deviated
quantities) from observed option price data. In a first part we review results concerning the
applicability of Tikhonov regularization to the inverse problem of option pricing, which is
2concerned with the identification of the squared volatility function a(t) :=σ (t) (t∈ [0,T]).
After that we concentrate on the identification of an antiderivativeS ofa, which leads to an
operator equation with a Nemytskii operator generated by a monotone function. Applying
results of Chapter 5 we prove well-posedness in aC-space setting and discuss ill-conditioning
effects which lead to strongly oscillating solutions. Therefore, we discuss the applicability
of monotonicity information for stabilising the solution process. As a result we propose a
numerically effective algorithm for the computation of a strictly monotonically increasing
solution and illustrate its performance by means of a numerical case study.
Finally, Chapter 7 is concerned with the estimation of the real-valued parameters in the
considered generalised Ornstein-Uhlenbeck model. Replacing the unobservable state process
by a scaled version we can reduce the number of parameters. After deriving the state space
representation of the considered model we use the Kalman filter to obtain certain condi-
tional expectation vectors. Using these quantities we can set up the log-likelihood function.
Furthermore, we present a short numerical case study and discuss briefly chances and lim-
itations of the method. In this context the effects of small noise in the volatility input are
investigated.Chapter 2
Stochastic Preliminaries
This chapter is intended as survey over a wide range of stochastic topics which will be used
throughout the thesis. We start with an elementary inequality and properties of Gaussian
random variables. After that we move on to the central limit theorem, which is used in the
proof of Theorem 4.3.2. Next, we introduce random processes, especially the Wiener process
with respect to a filtration. In order to be brief we abstain from defining the Stochastic
Itô integral and refer simply to the literature, for instance [30]. Nevertheless, we define the
Stochastic differential and formulate two versions of the Itô formula which are used quite
frequently in Chapter 4.
Lemma 2.1.1
Let ξ (i = 1,...,n) be random variables with finite variances. Then it holdsi !
n nX X
2 2
D ξ ≤n D ξ. (2.1)i i
i=1 i=1
Proof: Cauchy-Schwartz inequality gives ! ! !2n n nX X X1 1
2(ξ −Eξ ) ≤ (ξ −Eξ ) .i i i i2n n
i=1 i=1 i=1
Hence we have ! ! !2n n n nX X X X1 1 1 12 2 2
D ξ =E (ξ −Eξ ) ≤E (ξ −Eξ ) = D ξ .i i i i i i2n n n n
i=1 i=1 i=1 i=1
2Multiplying both sides by n gives the assertion.
Properties of Gaussian random variables
Gaussian random variables play an extremely important role in probability theory and con-
sequently in mathematical finance. One reason for this is that they have properties that
10